A Priori Error Estimates for a Finite Element Discretization of Parabolic Optimization Problems with Pointwise Constraints in Time on Mean Values of the Gradient of the State

2015 ◽  
Vol 53 (2) ◽  
pp. 745-770 ◽  
Author(s):  
Francesco Ludovici ◽  
Winnifried Wollner
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Optimization Problems ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Mean Values ◽  
Priori Error Estimates ◽  
Pointwise Constraints

scholarly journals A priori error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

2021 ◽  
Author(s):  
Marita Holtmannspötter
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
A Priori ◽  
Damage Model ◽  
Space Time ◽  
A Priori Error Estimates ◽  
Element Approximation ◽  
Element Discretization ◽  
Priori Error Estimates ◽  
Almost All

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.

scholarly journals A priori error estimates for a linearized fracture control problem

2020 ◽  
Author(s):  
Masoumeh Mohammadi ◽  
Winnifried Wollner
Keyword(s):  
Control Problem ◽  
Error Estimates ◽  
Optimization Problems ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Adjoint Variables ◽  
Conforming Finite Element Method ◽  
Priori Error Estimates ◽  
Fracture Control ◽  
Control State

Abstract A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived.

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